Equation of a straight line
The general equation appears as \(Ax + By + C = 0\).
However to build up an equation use \(y - b = m(x - a)\) where \(m\) is the gradient and \((a,b)\) is a point on the line.
Example 1
Find the equation of the line with gradient 3, passing through \((4,1)\).
Solution
Using \(y - b = m(x - a)\) with \(m = 3\) and \((a,b) = (4,1)\), we get:
\(y - 1 = 3(x - 4)\)
\(y - 1 = 3x - 12\)
\(y = 3x - 11\)
\(3x-y-11=0\)
To identify features compare with the form \(y = mx + c\) where \(m\) is the gradient and \((0,c)\) is the y-intercept.
Example 2
Find the gradient of the line with equation \(2x + 5y - 6 = 0\)
Solution
Rearrange this in the form \(y = mx + c\) to get:
\(2x + 5y - 6 = 0\)
\(5y = - 2x + 6\)
\(y = - \frac{2}{5}x + \frac{6}{5}\)
\(gradient = - \frac{2}{5}\)